It is hardly just Reagan's or Obama's recovery- too many other variables in play. I think the Fed gets a lot of the credit for this one and while it isn't the greatest, it is also not bad and could be far worse overall. If only the Fed could do more, but I think political nutcasery on the Hill has held them back.
There are many reasons for labor force participation rate changes. You can't just look at the number and blame it on the Presidency. Like I said, systemically Obama's recession was probably worse given the state of the financial system and Fed rates that were already relatively low. He is also President during a time when automation is more and more a substitute for lower level labor and where global competition is greater. A lot of capital that may have added jobs in Reagan's days may be adding computers or machines today. And that is no fault of Obama's, the Democrats, Republicans, or anyone else.
Then there is the whole issue of the Federal Reserve- the side that many political discussions seem to leave out for whatever reason. Amongst macroeconomists, the effectiveness of fiscal policy vs. monetary policy in spurring a recovery is still highly contentious. I think Volcker and Bernanke both did great jobs doing what they had to do in their tenures, by the way. I preferred some of Reagan's fiscal measures to Obama's too (not a fan of the stimulus as it was done- would've preferred tax cuts), but they presided over different times. Overall, it could be a lot, lot worse (imagine one of the Tea Party folks trying to slash spending across the board right now or in the last few years and actually having enough power to do so).
I think Greenspan was more ideological than some other chairmen of the Fed in how he saw markets and banking. He seemed to have a lot more faith in the private sector even when the numerical risks were partly in front of him. Might be due to his relative lack of an academic background. Reminds me, I need to check out his new book.